Ohhhhh! I think I could have some fun with this.
Grass - Metal
If you think of 'grass' as representing nature and 'metal' as representing technology it kind of makes sense. Basically a nature versus technology type of deal.
Fire - Water
This is a pretty common train of thought. It applies to most classical element systems throughout mythology.
Water - Fire
Electric - Magnet
While certainly related I don't know if it's fair to say electricity and magnetism are the same thing. However magnets can be used to generate an electric current and an electric current can be used to generate magnetic waves. They aren't opposites in the sense that they foil each other but they kind of counter one another in a way.
Bug - Cell
This one certainly is a tough one. Since it is really more of description than an element I'm inclined to go with some other form of description or classification. I think cells work nicely because they do contrast with bugs (ignore that bugs are made of cells) fairly well. Bugs have exoskeletons while cells have membranes. Bugs typically live in dry environments while cells typically inhabit moist ones. Bugs are complex organisms while cells are simplistic (To a degree, this is actually subjective.)
Flying - Ground
Since I typically view 'flying' as a 'wind' type more than a 'bird' type I would go with this one.
Ground - Flying
Normal - Normal
Who says an inverse can't be itself? Mathematics would beg to differ. For what 'Normal' is supposed to be I think it makes sense for normal to be an opposite to itself.
Ghost - Fairy
This one would vary greatly depending on which definitions and lore on ghosts and fairies you are drawing from. For me, I see fairies as being spirits of nature and ghosts as being spirits of civilization. It goes back, kind of, to nature vs technology but not really. Essentially I tend to view ghosts as being the product of people in some shape or form and faeries as being the product of nature.
Fighting - Peace
Fighting is one of those elements that really doesn't seem like an 'element' and is more like a temperament. It would make sense for a fighting type to thus be a piece type. It could be some kind of Pokemon based more on support than damage.
Psychic - Dull
The main concept behind the psychic type is having immense intelligence that allows you to essentially manipulate the world on a higher level of understanding than your average person can perceive. So the opposite of a psychic could be either 'normal' or 'dull.'
Rock - Paper
This one is mostly a pun on Rock, Paper, Scissors. It also kind of works on a general level. Rocks are hard, paper is soft. Rocks are rigid, paper is flexible.
Steel - Grass
Poison - Cure
Just a little bit more specific than 'Medical.'
Dark - Justice
In Japan the 'Dark' type is called 'Evil.' Good sounded kind of lame to me so I went with justice. They could have a hero motif.
Dragon - Serpent
Dragons typically have a very rich and regal lore. They are viewed as celestial in nature, to a degree. Serpents on the other hand are frequently evil creatures that have been cast down from the heavens. Yeah, Christianity definitely puts that out there but its a common element in mythology in general. There's also the aspect of dragons being capable of flying while serpents are stuck to the ground.
Ice - Steam
This one is one I'm not super happy with :/ .
Ice is just water in a different state of matter. I almost don't feel like it should be a type on that principle alone. But since ice is water as a solid I decided to make its inverse water as a gas.
Fairy - Ghost
EDIT:
A few other possibilities:
Electric - Ground
Since the process of getting something to lose some of its charge is by grounding it. (With the ground being about the best option for discharging electrons).
Flying - Rock
Again, using 'flying' as 'wind' if we thought of rock as being 'earth' it could fit in with the classic Greek elements.
